Pramana – J. Phys. (2020) 94:127 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-020-01990-1

Effect of suction/blowing on heat-absorbing unsteady radiativeCasson fluid past a semi-infinite flat plate with conjugate heatingand inclined magnetic field

R MAHATO and M DAS ∗

Department of Mathematics, School of Applied Sciences, KIIT Deemed to be University,Bhubaneswar 751 024, India∗Corresponding author. E-mail: mrutyunjay0307@gmail.com; mdasfma@kiit.ac.in

MS received 10 January 2020; revised 17 April 2020; accepted 13 May 2020

Abstract. This article studies the effect of suction/blowing, inclined magnetic field, and chemical reaction onheat-absorbing unsteady radiative Casson fluid past a semi-infinite flat plate in porous medium incorporating theoscillatory plate movement as a linear combination of ‘cosine’ and ‘sine’ functions in time. Further, the mass andheat transfer characteristics are examined under the influence of conjugate mass and heat transfer phenomena at theboundary. The governing equations of the model, viz. the energy, mass transfer, and momentum, are transformed intothe non-dimensional form adopting suitable non-dimensional variables and parameters. The exact analytic solutionsof the model for species concentration and fluid temperature are obtained using Laplace transform technique whilst,the solution for the fluid velocity has been obtained numerically with the help of the INVLAP routine of MATLAB.The expressions for fluid temperature, species concentration, and velocity are obtained and studied graphicallyfor various physical parameters influencing the fluid flow model taking into account the case of both suction andblowing. Further, the solutions when the Casson fluid parameter α → ∞ are also obtained as special cases. Resultsfor the skin friction coefficient, Sherwood number, and Nusselt number are numerically calculated and put in tabularform. An increment for the inclination angle of the magnetic field enhances the fluid velocity while it has a reverseeffect on skin friction. Increasing the Schmidt number Sc leads to a reduction in fluid concentration and increasingthe value of thermal radiation accelerates fluid temperature. This fluid flow model has several industrial applicationsin the field of chemical, polymer, medical sciences, etc.

Keywords. Suction; blowing; porous medium; Casson fluid; conjugate heat and mass transfer; inclined magneticfield; thermal radiation.

PACS Nos 47.10.ab; 47.11.−J; 47.35.Tv; 47.70.−n; 47.70.Fw

1. Introduction

Over the past several decades, tremendous effort hasbeen devoted to study the physical behaviour and prop-erties of non-Newtonian fluids. Non-Newtonian fluiddoes not obey Newton’s law of viscosity and therefore,the constant coefficient of viscosity cannot be defined.A thorough understanding of the chemically reactivenon-Newtonian flow behaviour in a porous medium andthe mass and heat transfer effect of the magnetic fieldhas considerable application in rheology, petroleumengineering, polymer industry, chemical engineering,cosmetic and pharmaceutical industries, for examplein the manufacture of cleansers, toothpaste, gel, sham-poo, honey, starch suspension, paint etc. Additionally,

thermal radiative, chemically reactive, hydromagnetic,electrically conducting, incompressible and viscousfluid flow models are finding their applications in MHDgenerator, MHD pump, geothermal energy extraction,plasma studies, and cooling of nuclear reactors. Casson[1] introduced the Casson model in 1957. The influenceof rotation and radiation on unsteady hydromagneticnatural convection flow of an electrically conductingincompressible viscous fluid crossing an impulsivelymovable vertical plate via a porous medium along withthe inclined magnetic field was described by Sandeepand Sugunamma [2]. Ramandevi et al [3] discussed theflow of two distinct MHD non-Newtonian fluids with anew theory for heat flux known as Cattaneo–Christovtheory. This model presented the influence of thermal

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127 Page 2 of 16 Pramana – J. Phys. (2020) 94:127

relaxation time on Casson fluid. Reddy et al [4] inves-tigated the heat transfer effect and viscous dissipationon unsteady MHD free convective, electrically conduc-tive Casson fluid across an oscillating vertical plate. Thesolutions of non-dimensional boundary layer partial dif-ferential equations (PDEs) of the flow are obtained byusing the finite element method (FEM). The impactof viscous dissipation on MHD micropolar fluid flowover a stretching thin surface with modified Fourier’smodel was discussed by Kumar et al [5]. Mustafa etal [6] discussed the influence of heat transfer on theunsteady flow of a Casson fluid past a moving flat platealong a parallel free stream. Effect of chemical reaction,mass and heat transfer, and cross-diffusion on nonlin-ear convective flow of Walters-B and Casson nanofluidscrossed a variable thickness sheet was studied by Lak-shmi et al [7]. Akbar [8] studied the peristaltic Cassonfluid flow through an asymmetric channel with mag-netic effect. Khalid et al [9] analytically presented thesolutions of MHD unsteady free convective Casson fluidflow on an oscillatory vertical porous plate with constantwall temperature. Kataria and Patel [10] investigatedthe unsteady hydromagnetic flow of Casson fluid overan exponentially accelerated vertical porous plate. Theeffect of suction, Dufour, thermophoresis, viscosity andtemperature-dependent, thermal conductivity nth-orderchemical reaction of a Casson fluid including the perme-able flat plate was studied by Animasaun [11]. Rao et al[12] discussed the effect of heat transfer on thermal andhydromagnetic slip condition of Casson fluid flow over asemi-infinite vertical plate. The stagnation point flow ofMHD micropolar fluid over a vertical stretching surfacewith thermal radiation and velocity slip was analysedby Kumar et al [13]. The non-aligned stagnation pointflow of radiative MHD micropolar fluid over a convec-tive stretched surface was presented by Kumar et al [14].Nadeen et al [15] investigated the influence of heat trans-fer on oblique stagnation point flow of a non-Newtonian(Casson fluid) fluid along a stretching surface usinghomotopy analysis method (HAM). Simultaneous solu-tions for first- and second-order velocity slips on MHDmicropolar fluid flow across a convective stretching sur-face under the influence of variable heat source/sinkand Lorentz force was presented by Kumar et al [16].The three-dimensional MHD Carreau fluid flow over astretching surface with improved Fourier’s model wasstudied by Ramadevi et al [17]. Viscous and Joule heat-ing, mass transfer and inclined magnetic field effectwith thermal radiative and chemically reactive Cassonfluid flow towards a stretching surface was investigatedby Gopal et al [18]. The Cattaneo–Christov heat fluxmodel for MHD micropolar fluid flow past a coagulatedsheet was studied by Kumar et al [19]. The mixed con-vective stagnation point flow of a radiative micropolar

nanofluid under the influence of thermophoresis, Brow-nian motion, viscous and Joule dissipation was studiedby Kumar et al [20].

In the last few years, many investigators are attractedto MHD viscous boundary layer flows via the porousmedium with thermal radiation due to its applicationsin geothermal energy extraction and cooling of nuclearreactor. Such cooling is known as transpiration cooling.Seth et al [21] presented a model of unsteady, natu-ral convective and hydromagnetic transient flow withramped wall temperature across an impulsively movingporous plate. Raptis [22] investigated the convective,electrically conducting, unsteady incompressible fluidflow along an infinite vertical porous plate in a two-dimensional case. Chamkha et al [23] studied the elec-trically conducting natural convective unsteady powerlaw fluid over a vertical non-Darcian porous plate underthe influence of homogeneous chemical reaction andnumerical solutions were obtained by the finite differ-ence method. Nandkeolyar and Das [24] investigatedthe influence of radiative heat transfer and natural con-vective MHD fluid flow via porous medium along withan inclined magnetic field and ramped temperature.Here a comparison was made for the obtained resultsin the case of an isothermal plate with ramped walltemperature. Laminar free convective, two-dimensionalunsteady fluid flow past an infinite vertical permeableplate of constant temperature was studied by Helmy[25]. Kim [26] studied the effect of variable suction andfree convective MHD flow through a movable semi-infinite vertical plate through a permeable medium.Makinde and Sibanda [27] discussed the heat transfercharacteristics of hydromagnetic mixed convective andsteady laminar flow over a vertical plate in a porousmedium. Chamkha [28] considered the MHD flow influ-enced by thermal radiation, heat sink or source andbuoyancy effects through an accelerating porous sur-face. MHD radiative micropolar fluid over a stretchingporous sheet with Joule heating and second-order veloc-ity slip was discussed by Kumar et al [29]. MHD Carreaufluid flow with Cattaneo–Christov heat flux over a melt-ing surface was analysed by Kumar et al [30]. Mustafa[31] studied the influence of viscous dissipation, freeconvection and thermal radiation on unsteady MHDflow past an infinite permeable vertical plate. Inclinedmagnetic field effect on viscous, incompressible, Cou-ette flow of a MHD fluid between two parallel platein the rotating system was investigated by Guria et al[32]. The influence of inclined magnetic field on elec-trically conducting, Hartmann, incompressible, viscousfluid flow of a rotating system was studied by Seth etal [33]. Samiulhaq et al [34] considered the viscous,incompressible, electrically conducting, mass and heattransport on MHD unsteady fluid flow past a vertical

Pramana – J. Phys. (2020) 94:127 Page 3 of 16 127

porous plate along with ramped temperature and thermaldiffusion. Suction or injection effect on free convectivemass and heat transfer MHD flow over a stretching sheetwith diffusion thermoeffects was studied by Afify [35].Nandkeolyar et al [36] presented a model of suctionor blowing effect on unsteady, viscous, free convectiveMHD, electrically conducting, heat radiating, chemi-cally reacting, incompressible fluid flow across a flatplate in the presence of ramped wall temperature. Heattransfer and thermal radiation effect on Casson fluid flowover a curved exponentially stretching surface with con-vective boundary condition was discussed by Kumar etal [37]. Nandkeolyar et al [38] investigated the heat andmass transfer effects of unsteady MHD, heat absorbing,electrically conducting, viscous, incompressible fluidpast a time-dependent infinite vertical flat plate. Theinfluence of heat source/sink on thin film flow of MHDradiative hybrid ferrofluid was discussed by Kumar etal [39]. They found that heat transfer rate is higher inhybrid ferrofluid than in ferrofluid.

In many materialistic cases, constant temperature failsto work. Consequently, nowadays, Newtonian heating isgiven more attention than the constant surface temper-ature where the rate of heat transfer from the boundingsurface having a finite heat capacity is proportional to thelocal surface temperature and which is generally namedas conjugate convective flow. The Newtonian heatingeffect is applicable in solar radiation, heat exchange,petroleum industry, conjugate heat transfer around fins,etc. The Newtonian heating effect on natural convectiveboundary layer flow through a vertical plate is inves-tigated by Merkin [40]. Hussanan et al [41] analysedthe exact solutions of heat transfer and Newtonian heat-ing effect on unsteady boundary layer flow of a Cassonfluid past an oscillating vertical plate. Das et al [42]discussed the influence of mass and heat transfer andNewtonian heating on unsteady MHD thermal radiativeand chemically reactive Casson fluid flow past a ver-tical plate. Laplace transform technique was employedto solve the governing equations. The effect of New-tonian heating on the natural convection MHD flowalong with mass diffusion was discussed analyticallyby Vieru et al [43]. Das et al [44] discussed the double-diffusive unsteady Casson fluid flow past a flat platewith an applied magnetic field and chemical reaction.The governing equations are solved numerically by theMATLAB program bvp4c. Ulah [45] studied the impactof Newtonian heating on MHD Casson fluid flow withslip condition over a nonlinearly permeable stretchingsheet. Here, Keller-box method was applied to solvethe governing equations. Khan et al [46] discussed thenatural convective and heat-generating MHD flow overa movable plate through a porous medium along withchemical reaction.

From the above discussions, it is very much clearthat the effect of suction/blowing and conjugate heat-ing on heat-absorbing, chemically reactive, and radiativeunsteady Casson fluid across an infinite flat plate inthe presence of an inclined magnetic field incorpo-rating the plate movement as a linear combination of‘cosine’ and ‘sine’ functions in time t is not being dis-cussed so far. Consequently, the objective of the presentanalysis is to make an attempt to analyse the thingsthrough graphs and tables. The governing partial dif-ferential equations are converted to non-dimensionalforms by adopting appropriate non-dimensional vari-ables and parameters that are then solved analyticallyby the Laplace transform technique. The expressions forfluid velocity, fluid temperature, and fluid concentrationare obtained and studied graphically for various physicalparameters influencing the fluid flow model taking intoaccount the cases of both suction and blowing. Further,the numerical results for skin friction, Nusselt num-ber, and Sherwood number are computed and presentedin tabular forms. It is expected that findings from thepresent fluid flow model will be applicable in medical,engineering, and various industrial processes.

2. Mathematical formulation

Here we assume an electrically conducting, heat-absorbing, heat and mass transport on an unsteadyfree convective, incompressible, viscous, hydromag-netic, and radiative Casson fluid past a semi-infinitevertical flat plate through porous medium includingthe influence of chemical reaction. Suction/blowingthrough a flat porous plate is also considered. Let x ′-axis be along the plate in the upward direction, y′-axisis normal to it and z′-axis is normal to x ′y′ plane. Thephysical representation of the problem is given in fig-ure 1. The fluid is permeated by a uniform magneticfield B0, making an angle θ with the positive direc-tion of y′-axis. Initially, both the plate and the fluid areat static condition having constant concentration C ′∞and temperature T ′∞. When t ′ > 0 the plate starts tooscillate against the gravitational field in its own planehaving velocity u′ = U0 cos(ω′t ′)+U0 sin(ω′t ′). At thesame time, mass transfer and heat transfer analyses aredone in the presence of conjugate mass and heat trans-fer respectively. In x ′ and z′ directions, the plate is ofinfinite length and except pressure all electrically non-conducting physical quantities are functions of only y′and t ′. The effect of the magnetic field initiated by themotion of fluid is insignificant compared to the appliedmagnetic field [47] so that the magnetic field strengthB ≡ (B0 sin θ, B0 cos θ, 0). The impact of the mag-netic field polarisation is insignificant as no external

127 Page 4 of 16 Pramana – J. Phys. (2020) 94:127

Figure 1. Flow geometry.

electric field is applied [48]; that is, E = (0, 0, 0). Thefluid flow is initiated by the impulsive movement of theplate in the x ′ direction, and in y direction there is aconstant flow due to the pores in the plate so that thevelocity vector is q = (u, −v0, 0).

The rheological equation of an isotropic and incom-pressible flow of the Casson fluid can be written as [1,41]

τ = τ0 + μα∗, (1)

or equivalently,

τi j =

⎧⎪⎪⎨

⎪⎪⎩

2

(

μB + py√2π

)

ei j , π > πc,

2

(

μB + py√2πc

)

ei j , π < πc,

(2)

where π = ei j ei j and ei j is the (i, j)th componentof the deformation rate, π is the product deformationrate itself, πc is the critical value based on the non-Newtonian model, and py is the yield stress of the fluid.

Under the above-made assumptions and conditions,the velocity, temperature, and concentration are func-tions of (y, t) only. Hence, the governing equations ofthe fluid flow model will be reduced to the followingform:

∂u′

∂t ′− v0

∂u′

∂ y′ = ν

(

1 + 1

α

)∂2u′

∂ y′2 − σ B20 cos2 θ

ρu′

− ν

K ′ u′ + gβT (T ′ − T ′∞)

+ gβC (C ′ − C ′∞), (3)

∂T ′

∂t ′− v0

∂T ′

∂y′ = k

ρcp

∂2T ′

∂y′2

− Q0

ρcp(T ′ − T ′∞) − 1

ρcp

∂q ′r

∂y′ , (4)

∂C ′

∂t ′− v0

∂C ′

∂y′ = D∂2C ′

∂y′2 − kr (C′ − C ′∞). (5)

The boundary and initial conditions are

u′ = 0, T ′ = T ′∞, C ′ = C ′∞,

for y′ ≥ 0 and t ′ ≤ 0,

u′ = U0 cos(ω′t ′) +U0 sin(ω′t ′), ∂T ′

∂y′ = −h1T′,

∂C ′

∂y′ = −h2C′ at y′ = 0 for t ′ > 0,

u′ → 0, T ′ → T ′∞, C ′ → C ′∞as y′ → ∞ for t ′ > 0.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(6)

Here, with the assumption that the thermal conductivityof the fluid is linear with temperature, the radiative fluxvector q ′

r under Rosseland approximation [49] becomes

q ′r = −4σ ∗

3k∗∂T ′4

∂y′ , (7)

where we assume that the differences in temperaturewithin the flow are such that T ′4 can be expressed as alinear combination of the temperature. The expansionof T ′4 into the Taylor series about T ′∞, after ignoringsecond- and higher-order components in (T ′ − T ′∞), is

T ′4 ∼= 4T ′3∞T ′ − 3T ′4∞. (8)

Using eqs (7) and (8), in eq. (4), we obtain∂T ′

∂t ′− v0

∂T ′

∂y′ = k

ρcp

∂2T ′

∂y2 − Q0

ρcp(T ′ − T ′∞)

+ 1

ρcp

16σ ∗T ′3∞3k∗

∂2T ′

∂y2 . (9)

Q0 is the heat absorption coefficient. The effect ofradiation in the thermal boundary layer equation (9) isanalysed in [50,51]. To reduce the governing equations(3), (5) and (9), into non-dimensional form, we obtainthe following non-dimensional parameters and quanti-ties:

y = y′U0

ν, u = u′/U0, t = t ′

U 20

ν,

T = (T ′ − T ′∞)/T ′∞,

C = (C ′ − C ′∞)/C ′∞, S = v0/U0, Kr = krν

U 20

,

M = σ B20ν/ρU 2

0 , Gr = gβT νT ′∞/U30 ,

Gm = gβCνC ′∞/U30 ,

Pr = ρνcp/k, ω = ω′ ν

U 20

, K1 = k′U 20

ν2 ,

Sc = ν

D, φ = νQ0

ρcpU 20

and N = 16σ ∗T ′3∞/3kk∗.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10)

Pramana – J. Phys. (2020) 94:127 Page 5 of 16 127

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

C(y

,t)

Suction(S = 1)Blowing(S = −1)

Sc = 0.5, 1, 1.5

(a)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

y

C(y

,t)

Suction(S = 1)

Blowing(S = −1)

Kr = 0.5,1,1.5

(b)

Figure 2. The influence of variation of (a) Schmidt number number Sc for γ2 = 0.5, Kr = 0.5 and (b) chemical reactionparameter Kr for γ2 = 1.5, Sc = 1 on the species concentration C(y, t).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

y

C(y

,t)

Suction(S = 1)

Blowing(S = −1)

t = 0.3, 0.5, 0.7

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

C(y

,t)

S = −1, −2, −3

S = 1, 2, 3

(b)

Figure 3. The influence of variation of (a) time t and (b) S on the fluid concentration C(y, t) for Sc = 1, Kr = 0.5, γ2 = 0.5,t = 0.7.

Converting into non-dimensional form the governingequations (3), (9), and (5), become

∂u

∂t− S

∂u

∂y=

(

1 + 1

α

)∂2u

∂y2

− M cos2 θu − u

K1+ GrT

+GmC, (11)

∂T

∂t− S

∂T

∂y= (1 + N )

Pr

∂2T

∂y2 − φT, (12)

∂C

∂t− S

∂C

∂y= 1

Sc

∂2C

∂y2 − KrC. (13)

The non-dimensional form of the corresponding ini-tial and boundary conditions (6) becomes

u = 0, T = 0, C = 0 for y ≥ 0 and t ≤ 0,

u = cos ωt + sin ωt,∂T

∂y= −γ1(1 + T ),

∂C

∂y= −γ2(1 + C) at y = 0 for t > 0,

u → 0, T → 0, C → 0 as y → ∞ for t > 0.

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(14)

127 Page 6 of 16 Pramana – J. Phys. (2020) 94:127

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

C(y

,t)

Suction(S = 1)

Blowing(S = −1)

γ2 = 0.3,0.5,0.7

(a)

0 1 2 3 4 5 6 70

2

4

6

8

10

12

y

T(y,

t)

Suction(S = 1)

Blowing(S = −1)

γ1=0.3, 0.5, 0.7

(b)

Figure 4. The influence of variation of conjugate parameter for (a) mass transfer γ2 and (b) heat transfer γ1 on the fluidconcentration C(y, t) and temperature T (y, t) for t = 0.7, Kr = 0.5, Pr = 0.71.

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

y

T(y,

t)

Suction(S = 1)

Blowing(S = −1)

t = 0.3, 0.5, 0.7

(a)

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

y

T(y,

t)

Pr = 0.5, 0.71, 1

Blowing(S = −1)Suction(S = 1)

(b)

Figure 5. The influence of variation of (a) time t and (b) Prandtl number Pr on the fluid temperature T (y, t) for t = 0.7,Pr = 0.71, γ1 = 0.5, N = 3.

Here,

γ1 = h1ν

U0and γ2 = h2

ν

U0. (15)

It is noted that in eq. (14), T = 0 and C = 0, whenγ1 = 0 and γ2 = 0 for t > 0. From this we can phys-ically interpret that there is no heat and mass transferfrom the plate.

3. Solution of the problem

The system of coupled equations (11)–(13) of the modelsubject to the boundary and initial conditions as definedin (14) are solved analytically with the help of theLaplace transform technique and the expressions for

T (y, s), C(y, s), and U (y, s) are presented respectively,as

T (y, s) = −γ1

s(λ1 + γ1)eyλ1, (16)

C(y, s) = −γ2

s(λ2 + γ2)eyλ2, (17)

U (y, s) =(

s

s2 + ω2 + ω

s2 + ω2 − α1

D1− α2

D2

)

eyλ3

+ α1

D1eyλ1 + α2

D2eyλ2, (18)

where T (y, s), C(y, s), U (y, s) are the transforms ofT (y, t), C(y, t), and u(y, t) respectively, s > 0 is theLaplace transform parameter, and

Pramana – J. Phys. (2020) 94:127 Page 7 of 16 127

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

y

T(y,

t)

Suction(S = 1)

Blowing(S = −1)

N = 1, 3, 5

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 104

y

T(y,

t)

φ = 1, 3, 5

Suction(S = 1)

Blowing(S = −1)

(b)

Figure 6. The influence of variation of (a) thermal radiation N and (b) heat absorption parameter φ on the fluid temperatureT (y, t) for t = 0.7, Pr = 0.71, γ1 = 0.5, N = 3, φ = 1.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

y

T(y,

t)

S=−1,−2,−3

S=1,2,3

(a)

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)S = 0.1S = 0.2S = 0.3S = −0.1S = −0.2S = −0.3

(b)

Figure 7. The influence of variation of (a) suction S > 0 and blowing S < 0 on the fluid temperature T (y, t) for t = 0.7,Pr = 0.71 and (b) suction S > 0 and blowing S < 0 on the fluid velocity for α = 1, γ1 = 0.3, γ2 = 0.3, Sc = 2.

λ1 = 0.5

{

− S

(Pr

1 + N

)

−√

S2

(Pr

1 + N

)2

+ 4

(Pr

1 + N

)

(s + φ)

}

,

λ2 = 0.5{−SSc −√S2Sc2 + 4Sc(Kr + s)},

λ3 = 0.5{−b′ −√b′2 + 4(a2 + a1s)},

D1 = λ12 + b′λ1 − (a2 + a1s),

D2 = λ22 + b′λ2 − (a2 + a1s),

a′ =(

1 + 1

α

)

, b′ = S

a′ ,

a1 = 1

a′ , a2 = Mcos2 θ + 1K1

a′ ,

α1 = Grγ1

a′s(λ1 + γ1)and α2 = Gmγ2

a′s(λ2 + γ2). (19)

The closed form solutions of eqs (16) and (17) areobtained using the inverse Laplace transform techniquesfor the fluid temperature T (y, t) and C(y, t) respec-tively. But it is difficult to get the exact inversion of (18)for the fluid velocity u(y, t). So it has been accessednumerically using the INVLAP routine in MATLAB.

127 Page 8 of 16 Pramana – J. Phys. (2020) 94:127

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

y

u(y,

t)

Suction(S = 0.05)

Blowing(S = −0.05)

t=0.3, 0.5,0.7

(a)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

y

u(y,

t)

Suction(S = 0.05)Blowing(S = −0.05)

α = 1, 1.5, 2

(b)

Figure 8. The influence of variation of (a) time t and (b) Casson fluid parameter α on the fluid velocity u(y, t) for Gr = 3,Gm = 2, γ1 = 0.3, γ2 = 0.3, Sc = 0.05.

0 1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

1.2

1.4

y

u(y,

t)

Suction(S = 0.05)

Blowing(S = −0.05)

M=0.1, 0.2, 0.3

(a)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

y

u(y,

t)

Suction(S = 0.05)

Blowing(S = −0.05)

θ=π/6, π/4, π/3

(b)

Figure 9. The influence of variation of (a) magnetic parameter M and (b) angle of inclination of magnetic field θ on the fluidvelocity u(y, t) for Gr = 3, Gm = 2, γ1 = 0.3, γ2 = 0.3, Sc = 0.05.

T (y, t) = e−yb

⎡

⎣m1(m2 +

√da )

2β{e−y

√derfc(t1)

− ey√derfc(t2)} − eβt m1m2

β

×{e−ym2√aerfc(t3) − eym2

√aerfc(t4)}

⎤

⎦ ,

(20)

C(y, t) = e−yq

⎡

⎣n1(n2 +

√rp )

2δ{e−y

√rerfc(t5)

− ey√rerfc(t6)} − eδt n1n2

δ

×{e−yn2√perfc(t7) − eyn2

√perfc(t8)}

⎤

⎦ ,

(21)

Pramana – J. Phys. (2020) 94:127 Page 9 of 16 127

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

y

u(y,

t)

Pr=0.5, 0.71, 1

Suction(S = 1)Blowing(S = −1)

(a)

0 1 2 3 4 5 6 7 8 9 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

N = 1, 2, 3

Suction(S = 1)Blowing(S = −1)

(b)

Figure 10. The influence of variation of (a) Prandtl number Pr for γ1 = 0.3, γ2 = 0.3, Sc = 0.05 and (b) thermal radiationN for γ1 = 2, γ2 = 2, Sc = 2 on the fluid velocity u(y, t).

where ‘erfc’ is the complementary error function and

t1, t2 = y√a

2√t

∓√d

at; t3, t4 = y

√a

2√t

∓ m2√t;

t5, t6 = y√p

2√t

∓√

r

pt; t7, t8 = y

√p

2√t

∓ n2√t;

a = Pr

1 + N, b = Pr S

2(1 + N );

c = Pr2S2

4(1 + N )2 , d = c + aφ;

m1 = − γ1√a, m2 = − b√

a+ γ1√

a, β = m2

2 − d

a;

p = Sc, q = SSc

2, r = S2S2

c

4+ ScKr ;

n1 = − γ2√p, n2 = − q√

p+ γ2√

p, δ = n2

2 − r

p.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)

4. Nusselt number, Sherwood number, and skinfriction

4.1 Nusselt number

Nu denotes the Nusselt number and it describes theenhancement of heat transfer from the plate. Expres-sion of the Nusselt number in non-dimensional form ispresented by

Nu = − ν

U0(T ′ − T ′∞

)

(∂T ′

∂y′

)

y=0

= γ1

(

1 + 1

T (0, t)

)

= −m1√a

×

⎛

⎜⎜⎝1+ 1

m1(m2+√

da )

βerf(

√da t)− 2m1m2

βeβterf(m2

√t)

⎞

⎟⎟⎠.

(23)

4.2 Sherwood number

Sh denotes the Sherwood number, which is also knownas the mass transfer Nusselt number that measures therate of mass transfer near the plate. Expression for theSherwood number in non-dimensional form is presentedby

Sh = − ν

U0(C ′ − C ′∞)

(∂C ′

∂y′

)

y=0

= γ2

(

1 + 1

C(0, t)

)

= − n1√p

×

⎛

⎜⎜⎝1+ 1

n1(n2+√

rp )

δerf(

√rp t)− 2n1n2

δeδterf(n2

√t)

⎞

⎟⎟⎠.

(24)

4.3 Skin friction

Skin friction is a non-dimensional quantity. The skinfriction coefficient physically means the ratio betweenlocal shear stress to characteristic dynamic pressure.According to some researchers, the velocity gradientindicates the shear stress level both in laminar as well

127 Page 10 of 16 Pramana – J. Phys. (2020) 94:127

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

u(y,

t)

Suction(S = 0.05)

Blowing(S = −0.05)

K1=0.2, 0.3, 0.4

(a)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

Suction(S = 0.05)

Blowing(S = −0.05)

Sc = 0.5, 1, 1.5

(b)

Figure 11. The influence of variation of (a) permeability parameter K1 and (b) Schmidt number Sc on the fluid velocityu(y, t) for K1 = 0.2, γ1 = 0.3, γ2 = 0.3, Sc = 0.05.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

Blowing(S = −0.05)

Suction(S = 0.05)

t = 0.3, 0.5, 0.7

(a)

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

Pr = 0.5, 0.71, 1

Suction(S = 0.05)

Blowing(S = −0.05)

(b)

Figure 12. The influence of variation of (a) time t and (b) Prandtl number Pr on the fluid velocity u(y, t) in limiting casefor Pr = 0.71, Sc = 2, t = 0.7.

as turbulent flow and there are several methods to esti-mate the same for the respective flow conditions. Herethe expression for skin friction coefficient is presentedby

τ ′ = −(

1 + 1

α

) (∂ u

∂y

)

y=0

= − a′{ (

s

s2 + ω2 + ω

s2 + ω2 − α1

D1− α2

D2

)

λ3

+ α1

D1λ1 + α2

D2λ2

}

. (25)

5. Limiting cases

The non-Newtonian nature of the Casson fluid becomesNewtonian when the Casson parameter is very large, i.e.α → ∞. In this section we have attempted to presentthe fluid velocity u(y, t) of the model in the case of theNewtonian fluid.

Pramana – J. Phys. (2020) 94:127 Page 11 of 16 127

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

θ = π/6, π/4, π/3

Suction(S = 0.05)

Blowing(S = −0.05)

(a)

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

M = 0.1, 0.2, 0.3

Suction(S = 0.05)

Blowing(S = −0.05)

(b)

Figure 13. The influence of variation of (a) angle of inclination of the magnetic field θ and (b) magnetic parameter M on thefluid velocity u(y, t) in limiting case for Gr = 3, Gm = 2.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

Sc = 1, 1.5, 2

Suction(S = 0.05)

Blowing(S = −0.05)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

y

u(y,

t)

S=0.1S=0.2S=0.3S=−0.1S=−0.2S=−0.3

(b)

Figure 14. The influence of variation of (a) Schmidt number Sc and (b) suction (S > 0) and blowing (S < 0) on the fluidvelocity u(y, t) in limiting case for Gr = 3, Gm = 2.

5.1 Velocity in the case of Newtonian fluid

The obtained solution for velocity presented in eq. (18)for the Casson fluid turns out to be the correspondingsolution for the Newtonian fluid as α → ∞ in the fol-lowing form:

U (y, s) =(

s

s2 + ω2 + ω

s2 + ω2 − α′1

D′1

− α′2

D′2

)

eyλ′3

+ α′1

D′1

eyλ1 + α′2

D′2

eyλ2, (26)

where

α′1 = Grγ1

s(λ1 + γ1), α′

2 = Gmγ2

s(λ2 + γ2),

D′1 = λ1

2 + Sλ1 − (b1 + s),

D′2 = λ2

2 + Sλ2 − (b1 + s),

b1 = M cos2 θ + 1

K1,

λ′3 = 0.5{−S −

√S2 + 4(b1 + s)}.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(27)

127 Page 12 of 16 Pramana – J. Phys. (2020) 94:127

5.2 Skin friction in the case of Newtonian fluid

The corresponding shear stress near the plate for theNewtonian fluid can be presented in the followingform:

τ ′1 = −

(∂U

∂y

)

y=0= −

(s

s2 + ω2 + ω

s2 + ω2

− α′1

D′1

− α′2

D′2

)

λ′3 − α′

1

D′1λ1 − α′

2

D′2λ2. (28)

6. Results and discussions

The effect of suction/blowing and conjugate heating onheat absorbing and chemically reactive unsteady radia-tive Casson fluid past a permeable plate in the presenceof an applied inclined magnetic field considering theplate movement as a linear combination of ‘cosine’ and‘sine’ functions in time t has been studied. The exactanalytic solutions of the model for fluid temperature andconcentration are obtained whilst, the fluid velocity hasbeen treated numerically by the MATLAB programmeof INVLAP routine. The effect of various parameterson the Casson fluid temperature T (y, t), concentrationC(y, t), and velocity u(y, t) profiles for both suctionand blowing cases are displayed in figures 2–10. For theNewtonian fluid, i.e. when α → ∞ the velocity profilesare depicted in figures 11–14. The impact of variousparameters on the Nusselt number, Sherwood number,and skin friction are portrayed in tables 1–3 respectively.

Figures 2a and 11b portray the impact of Schmidtnumber Sc on concentration and velocity profiles of thefluid. Since Sc is the ratio of kinematic viscosity andmass diffusivity, increment in mass diffusivity reducesSc. So, the increasing trend in Sc reduces the speciesconcentration as well as fluid velocity. The impact ofthe chemical reaction parameter Kr on the concentrationprofile is depicted in figure 2b. We observe that increas-ing Kr reduces the concentration profile. The rate ofinterfacial mass transfer is accelerated by Kr , and sothe species concentration decreases with an enhancingvalue of Kr . Figures 3a, 5a and 8a display the concen-tration, temperature and velocity of the fluid influencedby the time t . Increasing in time t gives increasingeffect on concentration, temperature, and velocity andwe observe from the three figures that in the case ofblowing, fluid temperature, concentration, and veloc-ity are higher than in the case of suction. Influence ofsuction and blowing are shown in figures 3b, 7a and7b. We observe that concentration, temperature, andvelocity of the fluid decrease with the increasing suc-tion velocity while blowing gives the reverse effect.In a porous medium, the fluid through the boundarylayer can be controlled by continuous suction or blow-ing. The wall shear stress and the friction drag can bereduced by blowing. The action of suction reduces themomentum boundary layer, thermal boundary layer andconcentration layer thickness corresponding to the retar-dation effect of velocity, temperature, and concentrationprofile.

Figures 4a and 4b highlight the effect of conjugateheat and mass transfer parameter on concentration andtemperature profile. We observe that while increasing

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

T(y,

t)

Present work

A.Hussanan et al.(2014)

t=0.1, 0.2, 0.3

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

y

T(y,

t)

Present work

A.Hussanan et al.(2014)

Pr=0.15, 0.3, 0.5

(b)

Figure 15. Comparison of exact solution of temperature T for (a) time t for γ1 = 0.5 and Pr = 0.3 and (b) Prandtl numberPr for γ1 = 0.5 and t = 0.2.

Pramana – J. Phys. (2020) 94:127 Page 13 of 16 127

Table 1. Nusselt number.

t S Pr N φ γ1 Nu (Suction) Nu (Blowing)

0.3 – 0.71 3 1 0.5 0.90065019 0.802852760.5 – 0.71 3 1 0.5 0.78847861 0.690382860.7 – 0.71 3 1 0.5 0.72883814 0.632477290.7 1 0.71 3 1 0.5 0.72883814 –0.7 2 0.71 3 1 0.5 0.79684580 –0.7 3 0.71 3 1 0.5 0.88180714 –0.7 −1 0.71 3 1 0.5 – 0.632477290.7 −2 0.71 3 1 0.5 – 0.599372780.7 −3 0.71 3 1 0.5 – 0.573873530.7 – 0.5 3 1 0.5 0.63881752 0.581074320.7 – 0.71 3 1 0.5 0.72883814 0.632477290.7 – 1 3 1 0.5 0.84550858 0.695160140.7 – 0.71 1 1 0.5 1.00208519 0.771618900.7 – 0.71 3 1 0.5 0.72883814 0.632477290.7 – 0.71 5 1 0.5 0.62712616 0.574242260.7 – 0.71 3 1 0.5 0.72883814 0.632477290.7 – 0.71 3 3 0.5 0.73544219 0.636863260.7 – 0.71 3 5 0.5 0.74208750 0.641326440.7 – 0.71 3 1 0.3 0.66279154 0.542666800.7 – 0.71 3 1 0.5 0.72883814 0.632477290.7 – 0.71 3 1 0.7 0.82210603 0.75708048

Table 2. Sherwood number.

t S Sc Kr γ2 Sh (Suction) Sh (Blowing)

0.3 – 1 0.5 0.5 2.24017728 1.387657760.5 – 1 0.5 0.5 1.91155966 1.121386410.7 – 1 0.5 0.5 1.74698955 0.978414970.7 1 1 0.5 0.5 1.74698955 –0.7 2 1 0.5 0.5 3.39273323 –0.7 3 1 0.5 0.5 10.37989795 –0.7 −1 1 0.5 0.5 – 0.978414970.7 −2 1 0.5 0.5 – 0.817402540.7 −3 1 0.5 0.5 – 0.710195830.7 – 0.5 0.5 0.5 1.10688167 0.800741310.7 – 1 0.5 0.5 1.74698955 0.978414970.7 – 1.5 0.5 0.5 2.57634643 1.097527930.7 – 1 0.5 0.5 1.74698955 0.978414970.7 – 1 1 0.5 1.87329816 1.064505810.7 – 1 1.5 0.5 1.99905148 1.53971210.7 – 1 0.5 0.3 2.00251963 0.894542200.7 – 1 0.5 0.5 1.74698955 0.978414970.7 – 1 0.5 0.7 1.72183837 1.07299671

the conjugate mass transfer parameter γ2 and conjugateheat transfer the parameter γ1 increase the thicknessof concentration and thermal boundary layer and as aconsequence the species concentration and fluid temper-ature increase respectively. The impact of Pr on fluidtemperature and velocity is displayed in figures 5b and10a for Pr = 0.5 (mixture of noble gas), Pr = 0.71(air), and Pr = 1 (gas). By increasing Pr the fluid

viscosity increases and consequently the temperatureand velocity profile decrease. This happens because anincrease in Pr increases the viscosity of the fluid, and soit becomes thick, consequently leading to the reductionin the velocity and temperature of the fluid. Figures 6aand 10b show the impact of thermal radiation N ontemperature and velocity profiles. Increasing N leadsto an increase in the thermal boundary layer and soit gives an increasing effect in the fluid temperature.Thermal radiation gives an increasing effect on fluidvelocity. The effect of radiation in the thermal boundarylayer equation (9), equivalent to an increased thermaldiffusivity Pr/(1 + N ) in eq. (12), can be consideredas an effective Prandtl number which decreases as thevalue N = 16σ ∗T ′3∞/3kk∗ for given k and T∞ impliesa decrease in Rosseland radiation absorptivity k∗. Theexpression of q ′

r in eq. (7) is the divergence of radia-tive heat flux which increases as k∗ decreases and thusthe approximation, taken in eq. (8), enhances the rate ofradiative heat transfer of the fluid, in turn, an incrementin the temperature takes place. Therefore, our work isanalysed under the effect of thermal radiation.

The impact of the heat absorption parameter φ ontemperature is displayed in figure 6b. The heat absorp-tion parameter gives a decreasing effect on temperature.The increasing values of φ reduce the temperature of theplate which in turn reduces the temperature of the fluid.The impact of the Casson fluid parameter α on the fluidvelocity profile is exhibited in figure 8b. Increasing thevalue of α tends to reduce the boundary layer thicknessas well as the velocity. It is noted that the higher value ofα tends to show the linear relationship between viscos-ity and shear stress (i.e. Newtonian fluid). The influenceof magnetic parameter M on fluid velocity profile isplotted in figure 9a. By increasing the value of M thevelocity is found to increase initially but after a certaindistance from the plate, it decreases with M . Lorentzforce works in the reverse direction of the fluid whichgives retardation effect on fluid velocity.

Figure 9b shows that an increase in inclination angle θ

accelerates the fluid velocity. This implies that when theapplied magnetic field acts transversely to the fluid flow,then the capacity of Lorentz force is maximum. Theeffects of permeability parameter K1 on fluid velocityis discussed in figure 11b. Increasing K1 in the boundarylayer region increases the fluid velocity. This happensbecause an increase in K1 decreases the resistance ofthe porous medium which enhances the fluid flow. Fig-ures 12–14 highlight the velocity profile in the limitingcases. Figures 12a and 13a show that velocity increaseswith increasing value of time t and inclined angle θ .Figures 12b, 13b, and 14a highlight that the increasingvalues of Pr , M , and Sc reduce the velocity profile.Figure 14b displays that suction gives decreasing effect

127 Page 14 of 16 Pramana – J. Phys. (2020) 94:127

Table 3. Skin friction when γ1 = 0.5, γ2 = 0.5, Sc = 1, Kr = 0.5, φ = 1, ω = 0.5.

t S Pr α M N θ K1 Gr Gm τ (Suction) τ (Blowing)

0.7 – 0.71 1 1 3 π/4 0.2 3 20.3 – – – – – – – – – 3.29322137 3.338061390.5 – – – – – – – – – 6.10096059 6.496235420.7 – – – – – – – – – 13.16090641 14.77407668– 0.05 – – – – – – – – 13.16090641 –– 0.10 – – – – – – – – 12.43696389 –– 0.15 – – – – – – – – 11.76319521 –– −0.05 – – – – – – – – – 14.77407668– −0.10 – – – – – – – – – 15.67137437– −0.15 – – – – – – – – – 16.63502723– – 0.5 – – – – – – – 12.41409234 14.03780438– – 0.71 – – – – – – – 13.16090641 14.77407668– – 1 – – – – – – – 13.55114231 15.15789081– – – 1 – – – – – – 13.16090641 14.77407668– – – 1.5 – – – – – – 59.87569829 70.56992005– – – 2 – – – – – – 397.46172417 840.53968937– – – – 1 – – – – – 13.16090641 14.03780438– – – – 2 – – – – – 16.49519528 14.77407668– – – – 3 - – – – – 20.87240159 15.15789081– – – – – 1 – – – – 13.7952735 15.39741836– – – – – 2 – – – – 13.50175835 15.10936677– – – – – 3 – – – – 13.16090641 14.77407668– – – – – – π/6 – – – 14.71565489 16.59893552– – – – – – π/4 – – – 13.16090641 14.77407668– – – – – – π/3 – – – 11.80125844 13.18406234– – – – – – – 0.2 – – 13.16090641 14.77407668– – – – – – – 0.3 – – 6.73351086 7.32533573– – – – – – – 0.4 – – 5.12536154 5.50174471– – – – – – – – 1 – 13.58913414 15.18450527– – – – – – – – 3 – 13.16090641 14.77407668– – – – – – – – 5 – 12.73267867 14.36364808– – – – – – – – – 2 13.16090641 14.77407668– – – – – – – – – 4 24.75761329 28.02136865– – – – – – - – – 6 36.35432018 41.26866062

Table 4. Comparison of skin friction in limiting case, i.e., α → ∞, when M = 0,Gm = 0, S = 0, K1 → ∞, θ = 0, φ = 0.

t Pr Gr γ1 ωt Hussanan et al [41] Das et al [42] Present result

0.01 0.35 5 1 0 5.5818 5.5818 5.58120.02 0.35 5 1 0 3.8620 3.8620 3.86060.01 0.50 5 1 0 5.5956 5.5956 5.59560.01 0.35 10 1 0 5.5218 5.5218 5.52050.01 0.35 5 2 0 5.5027 5.5027 5.50110.01 0.35 5 1 π/2 4.4819 4.4819 4.4819

to the fluid velocity while blowing has reverse effect onfluid velocity.

To study the impact of several parameters on the phys-ical quantities of practical significance, for instance,Nusselt number, Sherwood number, and skin friction,we have calculated their values numerically and pre-sented them in tables 1–3. The Nusselt number gets

accelerated with increasing values of Pr , γ1, φ andit has a reverse effect with t and N for both blowingand suction cases. Also, an increasing suction veloc-ity leads to an increase in the rate of heat transfer whileblowing shows the reverse effect. The Sherwood numberincreases with Sc and Kr and decreases with t for bothsuction and blowing. Here we observe that Sh decreases

Pramana – J. Phys. (2020) 94:127 Page 15 of 16 127

with increasing values of γ2 for suction while in blow-ing, it has a reverse effect. Suction velocity leads to anincrease in Sh while blowing gives the reverse effect.The skin friction increases with t , α, M , Pr , Gm anddecreases with K1, θ , N , Gr for both suction and blow-ing. Again, we observe that the blowing acceleratesskin friction while suction leads to reduced skin fric-tion value.

7. Validation of results

We observe that figure 15 and table 4 display the valida-tion of our present work by comparing it with publishedliteratures. Figures 15a and 15b display the comparisonof temperature profiles T with varying t and Pr whenS = 0, N = 0, φ = 0 with Hussanan et al [41] andthe present results are quite identical. We also observefrom table 4 that in the limiting case (i.e. α → ∞) whenM = 0, Gm = 0, S = 0, K1 → ∞, θ = 0, and φ = 0our result are in good agreement with Hussanan et al[41] and Das et al [42], which justify our present work.

8. Conclusion

Here, the influence of suction/blowing and thermal radi-ation on heat-absorbing and chemically reactive Cassonfluid flow past a porous plate in the presence of theinclined magnetic field is presented. The fluid tem-perature and concentration were considered along withNewtonian heating. Here velocity was considered as alinear combination of sine and cosine functions and con-sequently, the velocity was found for an oscillating plate.Laplace transform technique was employed to solve thefluid temperature, concentration, and velocity. The cor-responding Nusselt number, Sherwood number and skinfriction were established in terms of error and comple-mentary error functions. The Nusselt number, Sherwoodnumber, and skin friction were calculated numericallyand portrayed in tabular format. The important findingsare as follows:

• A comparative study of suction and blowing is high-lighted on velocity, temperature, and concentrationprofiles.

• Time t leads to an increase in fluid velocity, tempera-ture, and concentration for both suction and blowingcases.

• Increase in time t tends to reduce Nusselt number andSherwood number while skin friction has a reverseeffect.

• The fluid velocity accelerates with an increment ofthe inclined angle of the magnetic field and also ithas reverse effect on skin friction.

• By increasing the magnetic field M , the fluid veloc-ity is found to increase initially but after a certaindistance from the plate, the fluid velocity decreases.

• Regarding Casson fluid parameter α → ∞, theobtained solution can tally with some well-knownsolution for Newtonian fluid.

• Suction velocity leads to an increase in both the Nus-selt number and Sherwood number while blowinghas a reverse effect on them.

Acknowledgements

The authors are thankful to the reviewers for theirviewpoints and constructive suggestions. Their valuableadvices make this paper more relevant.

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